New Upper Bound for Sums of Dilates

For λ ∈ Z, let λ · A = {λa : a ∈ A}. Suppose r, h ∈ Z are sufficiently large and comparable to each other. We prove that if |A + A| 6 K|A| and λ1, . . . , λh 6 2r, then |λ1 ·A+ . . .+ λh ·A| 6 K |A|. This improves upon a result of Bukh who shows that |λ1 ·A+ . . .+ λh ·A| 6 K|A|. Our main technique is to combine Bukh’s idea of considering the binary expansion of λi with a result on biclique decompositions of bipartite graphs.